For Nietzche, happiness is a complex thing. Different shades of it have different meanings and can lead us to pursue opposite agendas.
Cheer and gaiety, for instance, lead the opposite direction from contentment. One can will to power or lust after it, and those lead to real differences in strategy.
Your view is close to Plato's. One way of looking at Platonic ethics is that we will for happiness, and that leads us to good.
Schopenhauer took this on from a sort of half-appropriated Buddhist point of view and deconstructed it, deciding that instead, we will for life. Life may be more abundant when we are happy, but it may also be more abundant when we are suffering for a purpose, or when we are made deeply sad by overwhelming empathy. We see people live out those choices, forsaking a big part of their potential happiness for purpose or authenticity.
Nietzsche would like to have gone from there back toward Plato, as he pulled back from the over-sophistication of Schopenhauer and his ilk in general. But he saw how powerful forms of happiness can lead us to become less human. Contentment can lead one to become a slave to a master or to a 'herd'. Satisfied lust can draw one into a limited and constricting master role from which only evil derives. Both are happiness, nonetheless.
So he classified 'happinesses', in an attempt to look deeper into the aspects of happiness that elevate and those that degrade humanity. From there we get to the will to power.
One way to put it is that the happiness that elevates (joy) is special, and is about a particular kind of self-expression, the will to express your deepest will and see its effects. That 'art of self' would be good even without the accompanying happiness. As Schopenhauer had pointed out, we see people who live abundantly by making great offerings of suffering for their conscience or principles, like Jesus of Nazareth, and are fulfilled by it, even if they are in anguish. The right action without the payoff can still be a good life if it expresses one's deepest will and has a chance of changing things.
So joy generally accompanies the will to power, but that will would be there even without the accompanying joy. And if your true will stops leading toward joy, it can still drive you to great purpose. The joy is a clue, and a good guide in general, but it is not the real motivation.
You can relabel the sense of purpose one gets out of following one's will as just a deeper form of happiness. But it is an abstract one that most folks cannot identify with. Stretching the definition of the term that much makes the Platonic argument even less convincing.